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Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Tomoyuki Ichiba  and
Constantinos Kardaras
Tomoyuki Ichiba*
Affiliation:
University of California Santa Barbara
Constantinos Kardaras*
Affiliation:
Boston University
*
∗ Postal address: Department of Statistics and Applied Probability, University of California Santa Barbara, CA 93106, South Hall 5607A, USA. Email address: ichiba@pstat.ucsb.edu
∗∗ Postal address: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA. Email address: kardaras@bu.edu
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Abstract

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We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1 / √N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.

Keywords

First passage timeMonte Carlo density estimationone-dimensional diffusionthree-dimensional Brownian bridgerate function

MSC classification

Secondary: 65C05: Monte Carlo methods 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 699 - 712
Copyright
Copyright © Applied Probability Trust 2011 

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